Exploring the WKB Approach for High Mass Particles in Quantum Mechanics

[eljose]

let,s suppose we have a particle with mass $$m\rightarrow\infty$$ then my question is if would be fair to make the WKB approach by setting the solution of the Schroedinguer equation as $$\phi=e^{iS/\hbar}$$ wiht S hte classical action satisfying the equation:

$$(dS/sx)^{2}+2m(V(x)-E_{n})=0$$ with E_n the Energies of the system...

 

[ZapperZ]

How do you propose in handling an infinite wavevector k?

Zz.

 

[R0man]

The WKB approach is a powerful tool in quantum mechanics that allows us to approximate the solutions to the Schrödinger equation for high mass particles. This approach is based on the idea that for high mass particles, the de Broglie wavelength is very small and the wave function can be approximated as a rapidly oscillating phase factor multiplied by a slowly varying amplitude.

In this case, it would be fair to use the WKB approach for high mass particles by setting the solution of the Schrödinger equation as ϕ=e^iS/ℏ, where S is the classical action. This satisfies the equation (dS/dx)^2+2m(V(x)-E_n)=0, where E_n is the energy of the system. This equation is known as the Hamilton-Jacobi equation and it is the classical counterpart of the Schrödinger equation.

The WKB approach allows us to solve for the wave function in terms of the classical action and the energy of the system. This means that we can use classical mechanics to approximate the quantum mechanical behavior of high mass particles. This is a useful approach because classical mechanics is often easier to solve and understand compared to the complexities of quantum mechanics.

However, it is important to note that the WKB approach is only an approximation and it is not always accurate. It is most useful for systems with high mass particles and low potential energies. For systems with low mass particles and high potential energies, the WKB approach may not be a good approximation and other methods, such as perturbation theory, may be more useful.

In conclusion, the WKB approach is a powerful tool in quantum mechanics that allows us to approximate the solutions for high mass particles. It is based on the idea that for these particles, the wave function can be approximated as a rapidly oscillating phase factor multiplied by a slowly varying amplitude. However, it is important to keep in mind its limitations and use it appropriately for the specific system at hand.